Proof of Theorem 12

Theorem 12.Strong Propriety for P entails
that, for all b, b′ in P and
E in F, if b(E) > 0 and
b′ ≠ b(• |
E) then

GExpU, E(b(• |
E) |
b) < GExpU,
E(b′ |
b)

That is, if our epistemic disutility function satisfies
Strong Propriety for P, conditionalizing on a piece of
evidence E minimizes expected disutility by the lights of the
agent's original credence function b and in the presence of
E.

We use the notation ΣA to denote the sum
over v in V that make proposition A
true. And we use the notation Σ to denote the sum over all
v in V.

By Strong Propriety for P, we have

GExpU, ⊤(b(• |
E) |
b(• |
E)) < GExpU,
⊤(b′ |
b(• |
E))

for b′ ≠ b(• |
E). That is,
prior to any evidence, the conditionalized credence function
b(• |
E) expects itself to be better than it
expects any other credence function to be. Thus, we have

Σ b(v |
E)U(b(• |
E), v)
< Σ b(v |
E)U(b′, v)

by the definition of GExpU, ⊤. Thus,
since b(v |
E) = b(v &
E)/b(E), we have: b(v |
E) = 0, if v does not make E true; and
b(v |
E) =
b(v)/b(E) if v does make
E true. Substituting this into the previous inequality, we
get